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COMPARISON OF SEVERAL STRATEGIES IN THE ACTIVE STRUCTURAL ACOUSTIC CONTROL USING STRUCTURAL STRAIN MEASUREMENTS
Journal of Sound and
COMPARISON OF SEVERAL STRATEGIES IN THE ACTIVE STRUCTURAL ACOUSTIC CONTROL USING STRUCTURAL STRAIN MEASUREMENTS P. MASSON AND A. BERRY G.A.;.S, Mechanical Engineering Department, ;niversiteH de Sherbrooke, Sherbrooke, QueH bec, Canada J1K 2R1 (Received 21 January 1999, and in ,nal form 16 December 1999) Two optimized approaches are herein proposed to lighten the real-time computation of the radiated acoustic power from planar structures using discrete strain measurements. These approaches are compared to a reference approach involving "nite di!erences reconstruction of the displacement "eld from the strain "eld. The "rst optimized approach allows the direct calculation of the radiated acoustic power from the discrete strain "eld while the second optimized approach employs an accelerated evaluation of the radiated acoustic power based on a wavelet transform of the displacement "eld. The evaluation of the radiated acoustic power from discrete strain values is experimentally assessed using the reference "nite di!erences approach and both optimized approaches. Piezoelectric "lm (PVDF) strain sensors are used for this validation as well as a feedforward "ltered-X LMS controller. Details on the real-time implementation of the cost functions are presented, especially on the o!-line adaptation of the control "lter. Formal comparisons using acoustic and vibration measurements are presented to better illustrate the control performance and mechanism of the di!erent approaches. Experimental results serve to demonstrate the interest of using the optimized approaches as compared to the reference approach and it is shown that a signi"cant gain in computational burden can be obtained using the proposed optimized approaches without sacri"cing the control performance. 2000 Academic Press 1. INTRODUCTION
The development of high-performance dedicated signal processors has driven the interest for active noise control in recent years. These processors are now more compact and a variety of tools exists to easily implement numerical controllers on them. It is well known that passive methods involving foams, viscoelastic layers and other absorbing or dissipative materials cannot be used e$ciently to reduce low-frequency noise. To perform noise control at those frequencies, di!erent active control strategies have been proposed and can be distinguished as active noise control (ANC) [1] and active structural acoustic control (ASAC) [2]. While active noise control requires the use of actuators located in the acoustic "eld, i.e., loudspeakers, the active structural acoustic control allows for the integration of both the error sensors and the actuators in the structure itself, following the trend towards smart materials. Moreover, it has been shown that, most of the time, ASAC has a lower requirement on the number of secondary sources needed to perform the control as compared to ANC [1]. Among the active structural acoustic control strategies, global and local strategies can be distinguished. Local control refers to spatially localized control (zones of silence) while global control is associated with extended control e!ect using cost functions closely related 0022-460X/00/240707#20 $35.00/0
2000 Academic Press
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P. MASSON AND A. BERRY
to energy quantities [3]. In this respect, the information given by the error sensors was used in the de"nition of cost functions based, for example, either on the far"eld pressure in prescribed directions [4}6], the volume velocity [7, 8] or the radiated acoustic power [9, 10]. The latter appears to be very promising as it ensures global control and, moreover, if working in the wavenumber domain [11, 12], allows for proper concentration of the control e!ort in the supersonic region of the structural displacement wavenumber spectrum, where the radiating wavenumber components are located. The control strategy based on the radiated acoustic power and de"ned in the wavenumber domain "rst requires the wavenumber transformation of the displacement (or acceleration) "eld of the structure. The supersonic (radiating) components of the spectrum can then be identi"ed and minimized. For this purpose, as compared to spatially extended sensors [13, 14], discrete sensors are more appropriate as they allow to distinguish between the structural components on two-dimensional structures [4]. Moreover, following the trend towards smart materials, strain sensors appear to be well suited for integration into structures and materials. As they do not measure directly the displacement or acceleration of the structure, the use of these sensors implies an additional step for the estimation of the radiated acoustic power. Therefore, there is a need to reduce the computational burden associated with the use of such sensors. This paper is thus concerned with the experimental validation of two optimized approaches which are herein proposed to evaluate the radiated acoustic power from discrete strain measurements. The cost function is de"ned as global and is expressed as the radiated acoustic power in the wavenumber domain and using discrete strain sensors. The "rst optimized approach allows the calculation of the radiated acoustic power directly from the strain "eld, without requiring any reconstruction of the displacement "eld. This "rst approach is based on a double integration by parts of the radiated acoustic power integral to provide an estimate in terms of the wavenumber transform of the strain "eld. The second optimized approach is directed towards the acceleration of the radiated acoustic power calculation by using a wavelet transform of a reconstructed displacement "eld, thus replacing both the Fourier transform and the integration of the radiating components [15]. In this second approach, the wavelet transform is chosen because it can have e$cient restriction on the supersonic region using a single space-wavenumber transformation. It is known that the wavelet transform has the property to localize the energy both in a given region of the wavenumber spectrum and spatially on the structure. The approach presented in this paper thus bene"ts from the wavelet framework, even if the analysis is herein restricted to the wavenumber localization in order to simplify the illustration of the approach. The reference approach based on a "nite di!erences reconstruction of the displacement "eld is "rst brie#y presented. Both optimized approaches are then presented and their control performance is experimentally compared to the reference approach.
2. ACOUSTIC RADIATION IN THE WAVENUMBER DOMAIN
For the sake of illustration, the acoustic radiation from a ba%ed rectangular thin beam is considered in this work. The acoustic pressure p(M) radiated at a point M from a ba%ed beam can be expressed by the Rayleigh's integral: o p(M)" 2n
=G (P, t)
e\ IP dA, r
(1)
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Figure 1. Experimental set-up: the ba%ed beam.
G (P, t) is the beam transverse acceleration at where o is the mass density of the #uid, = a point P, k is the acoustic wavenumber in the #uid, r is the distance between P and M and A is the surface of the beam, as presented in Figure 1. Assuming harmonic and constant displacement along the y direction such that =(P, t)"w(x) e SR, the previous equation can be used to obtain a far"eld approximation of the radiated acoustic pressure [16]: u o e\ I0 p(R, , u)"! wJ (j, k), 2n R
(2)
where j"k sin h cos u,
k"k sin h sin u
(3, 4)
and where R, h and u are the spherical co-ordinates of the point M. wJ (j, k) is the double wavenumber transform of the beam displacement, analogous to the inverse Fourier transform de"ned, for a beam of "nite width assuming constant displacement along the y direction, as [7] JV
JW
\J \J w(x, y) e I V>I W dy dx
wJ (k , k )" V W
V
V
W
(5)
W
2 sin k l JV WW " w(x) e\ IVV dx k W \JV
(6)
2 sin k l W W wJ (k ), " V k W
(7)
where l is the half-length of the beam, l is the halfwidth of the beam and k and k are, V W V W respectively, the x and y components of the structural wavenumber.
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The acoustic power P radiated from the beam can be expressed as the integral over a hemisphere of the mean square of the sound pressure in the far "eld:
1 P" " p(R, h, u) " dS, 2o c 1
(8)
where S is the surface of the hemisphere and c is the speed of sound in the #uid. In order to exploit the physical mechanisms of structural radiation, the radiated acoustic power is expressed in the wavenumber domain [11, 17, 14]. The radiated acoustic power in the wavenumber domain is obtained by substituting equation (2) into equation (8), as presented in detail in reference [18]: o u I (I\H " wJ (j, k) " dk dj. P" 8n \I \(I\H (k!j!k
(9)
In equation (9), the integration is performed over the supersonic region of the structural wavenumber spectrum (i.e., the region where the acoustic wavenumber k is larger than the structural wavenumber given by (j#k), which shows that only supersonic structural waves radiate sound in the far "eld. The minimization of the displacement wavenumber spectrum (weighted by (1/(k!j!k) in the supersonic region will then minimize the radiated acoustic power. For a narrow beam (for which " k l " 1) and using equations (2) and (7), equation (9) WW reduces to the form o u l I P " W " wJ (j) " dj. DGLGRC 2n \I
(10)
The objective of the following control approaches is to derive cost functions from structural strain measurements which approximate the exact radiated acoustic power given by equation (10).
3. APPROACHES FOR USING STRAIN MEASUREMENTS IN THE EVALUATION OF THE RADIATED ACOUSTIC POWER
Three approaches are herein presented for using strain measurements in the evaluation of the radiated acoustic power, through equation (10). The "rst approach, based on a "nite di!erences scheme for the reconstruction of the displacement "eld from the strain "eld, is referred to as the reference approach because it involves fewer assumptions than other approaches. Two optimized approaches are also presented: the "rst one is based on rewriting equation (10) in terms of the wavenumber transform of the strain "eld; the second one is based on a wavelet approximation of the radiated acoustic power. In the following, it will be assumed that the Bernoulli thin beam conditions apply, i.e., the strain e (x) at the surface of the beam is related to the second space derivative, in that V direction, of the displacement "eld w(x): h *w(x) e (x)"! @ , V 2 *x where h is the thickness of the beam. @
(11)
ASAC STRATEGIES USING STRAIN MEASUREMENTS
3.1.
711
REFERENCE APPROACH: FINITE DIFFERENCES SCHEME
A "nite di!erences scheme can be used to reconstruct the displacement "eld w(x) from the discrete strain "eld *w(x)/*x known on a p equally spaced grid of points on the beam (x ) G (i"1, 2, 2 , p, where i"1 and i"p are on edges) [10]. This scheme however requires the knowledge of two of the boundary conditions of the beam. The central-di!erence scheme can be written in a matrix form as A
w "w , N N> N> N
(12)
where the subscripts indicate the dimensions of the matrices or vectors. In this equation, w is the vector of the measured strain and w is built from the displacement at each N N> measurement point w and from the slopes of the displacement "eld in the x direction G w ,*w(x )/*x on the edges such that H H w w w w " . N> $
(13)
w N w N Indeed, it is seen that equtaion (12) can be solved only if two components of w are N> known a priori. These two components can be chosen from the "rst two components and the last two components of this vector, representing boundary conditions on the displacement or the slope along the x"x and x"x edges. In the following, four physical N strain measurement points are used, in addition to two assumed strain values on the edges. Under these conditions and for a simply supported beam, taking into account the zero-strain condition (w "0) as well as the zero-displacement condition (w"0) on the edges, the "nite di!erences scheme reduces to !2Dx
2 !2
1 Dx
1
1 !2
1
1 !2
1
1 !2 2 2Dx
w 0 w w w w " , w w w w w 0
(14)
where Dx is the space interval between two consecutive points and w "*w(x )/*x is the I I strain measured in the x direction. The reconstructed displacement "eld obtained from the vector w , by solving equation (14), and from the zero-displacement assumption on the edges, is then transformed using a discretized version of equation (6) given by wJ (k )" w e IV VGDx V G and substituted in equation (10) to obtain the radiated acoustic power.
(15)
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In order to obey the Nyquist criterion, it is expected that two measurement points will be required per mechanical wavelength to obtain an adequate estimate of the radiated acoustic power. Therefore, four strain sensors along the beam, together with the two assumed zero-strain values at the beam edges, should allow an adequate estimate up to the third vibration mode. Spatial aliasing can impair the estimate of the radiated acoustic power as the precision obtained on the wavenumber components of the displacement "eld is directly related to the accuracy of the radiated acoustic power estimate. The use of zero padding can improve the accuracy of the estimate by providing a larger number of components in the evaluation of the radiation integral given in equation (10). Spatial aliasing was extensively studied before through numerical simulations [10].
3.2.
OPTIMIZED APPROACHES
3.2.1. Direct approach The "rst optimized approach aims at expressing the radiated acoustic power (equation (10)) directly in terms of the Fourier transform of the strain measurements. This approach is based on a double integration by parts of the wavenumber transform (equation (5)). The "rst integration by parts with respect to the x variable leads to the following result for the wavenumber transform: 1 wJ (k , k )" V W ik X
1 ? @ *w(x, y) w(x, y)e IV V>IW Wn ) i dl! e IVV>IW W dy dx, ik *x V \? \@
(16)
where i is the unit vector in the x direction, n is the unit outer vector normal to an edge and Z denotes the integration over the contour of the surface. The second integration by parts with respect to the x variable gives
1 1 w(x, y) e IV V>IW Wn ) i dl# wJ (k , k )" V W k ik X V
*w(x, y) e IV V>IW W n ) i dl *x
1 ! SI (k , k ) V W k V
(17)
with ?
@ *w(x, y) e IV V>IW W dy dx, *x \? \@
SI (k , k )" V W
(18)
where SI (k , k ) is the wavenumber transform of the strain "eld. Equation (17) can be V W rewritten as 1 wJ (k , k )" I ! SI (k , k ), V W G k V W V G
(19)
where I are the subsisting boundary terms associated with edges i"1, 2 parallel to the G y-axis. Preceding work has shown that, for a simply supported structure, the boundary terms can be neglected at low frequency without a!ecting signi"cantly the radiated acoustic
ASAC STRATEGIES USING STRAIN MEASUREMENTS
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power estimate [10]. It was observed that, for a given mode, the boundary term in equation (19) and the surface strain transform term in equation (18) are of the same order of magnitude, below the modal frequency. The importance of the surface strain-transform term increases slightly close to that frequency while its relative importance decreases rapidly above this frequency. The global e!ect is that, with an increase in frequency, more and more modes are considered in which the surface-strain transform term has little e!ect or, conversely, the boundary term has large relative e!ect. Therefore, for a simply supported beam and using four strain sensors, the following one-dimensional form of equation (19) can be used if k O0: V 1 wJ (k )+! w eGIVVG Dx, V G k V G
(20)
and if k "0, the following form is used [10]: V
x l G ! V w Dx. wJ (0)+ G 2 2 G
(21)
3.2.2. =avelet approach In the "nite di!erences approach, the radiated acoustic power was related to the integration, in the supersonic region of the wavenumber domain, of the squared norm of the spectral components (corresponding to energy density) of the displacement "eld, as it appears in equation (10). The second optimized approach uses a space-wavenumber wavelet transform to observe the energy localized in the supersonic region. This approach has been theoretically investigated before [15] and a condensed presentation of the approach is only given here. The wavelet transform of a function f (x) is de"ned as
1 x!b f (x) t (= f ) (b, a)" dx, R a (" a " \
(22)
where t(x) is called the mother wavelet from which all the other wavelets are generated using a translation parameter b and a dilation parameter a. The overbar indicates the complex conjugate. It can be shown [19] that this wavelet transform gives localized information on a function f (x) within the space and wavenumber windows which can be de"ned in the root-mean-square (r.m.s.) sense, consequently leading to a localization which is also expressed in the mean-square sense. The information is then localized with the wavelet transform within a space window given by [b#ax*!aD , b#ax*#aD ], R R
(23)
where x* is the center and D is the r.m.s. extent (or radius) of the mother wavelet. In R a similar way, the wavelet transform is seen to give localized information in the wavenumber domain within a wavenumber window given by
i* D i* D ! R, # R , a a a a
(24)
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P. MASSON AND A. BERRY
where tI (i) is the Fourier transform of the mother wavelet and, as before, i* is the center and D I is the r.m.s. bandwidth (or radius) of the Fourier transform of this mother wavelet. R The quantities D and D I also de"nes space and wavenumber resolutions, associated with R R a particular wavelet. These resolutions satisfy the uncertainty principle [19]: D D I * . R R
(25)
The squared norm of the Fourier transform given by " fI (i) " is known to lead to a continuous representation of the energy density distribution of f (x) in the wavenumber domain, known as a spectrogram. The energy density is thus obtained only as a function of i. To obtain an energy density distribution of the function in the space}wavenumber plane, a similar quantity can be de"ned in the case of the wavelet transform as a squared norm given by " (= f ) (b, a) " R
(26)
and this quantity can be referred to as a scalogram [20]. The resolution capabilities of the scalogram depend on the analyzed wavenumber, as presented before. The shape of the wavelet t and of its Fourier tI obviously determines a weighting for the localization obtained in both domains. Following equations (23) and (24), the energy is distributed into space and wavenumber windows in the mean-square sense depending on the parameters b and a. Considering the above, it is "rst proposed in this section that the squared norm of the wavelet transform of the beam-displacement "eld, " (= w) (b, a) ", can observe the R contribution of this local displacement "eld to the total vibratory energy [15]. It is next proposed that the quantity given by
2n (= w) (b, a) u (" a " R
(27)
can be used to represent the acoustic power radiated by the beam if the parameter a is chosen such that the wavelet transform gives localized energy information in the supersonic region of the wavenumber spectrum. In other words, the choice of the parameter a as well as the use of the multiplying factor u are such that equation (27) mimics the exact expression of the radiated power (10). It is further assumed that a single component in the b-axis is su$cient to represent the energy in the supersonic region. Therefore, the parameter b is set to 0, implying that the wavelet is always centered on the beam. In the following, due to its simplicity, the Morlet wavelet will be chosen as the mother wavelet even if, rigorously, this wavelet is only numerically admissible, i.e., only numerical reconstruction of f (x) from its wavelet transform is possible. The Morlet wavelet t (x) is % de"ned as the modulated Gaussian window is given by 1 t (x) " : e\ LV> e\V ?, % 2(na
(28)
where a will be chosen in order to meet special requirements, as shown later. The following centers and radii are easily obtained for the Morlet wavelet: x*"0,
D "(a, R
i*"!2n,
1 DI" . R 2(a
(29}32)
ASAC STRATEGIES USING STRAIN MEASUREMENTS
715
It has been shown [15] that the following relation between the parameter a and the angular frequency u allows the wavelet transform to localize the energy in the supersonic region of the wavenumber spectrum at frequency u : c i* a" , u c
(33)
where a parameter 0)5)c)1 is used to ensure that, considering the center speci"ed for the window, its width will not cause the window to exceed the supersonic region. Therefore, a design criterion for the Morlet wavelet can be obtained under these conditions: 1 c a" . 16n (1!c)
(34)
In the preceding, the wavelet approach was presented using one-wavelet transform. However, in order to cover completely both space and wavenumber domains, i.e., the entire beam and the entire supersonic region, a few wavelet transforms have to be used and thus, a few wavelets in the wavenumber domain, each of them having b"0 and each of them reconstructing the energy in a given subregion of the supersonic region, and then summing all these contributions. By doing so, the wavenumber window of each wavelet is of limited extent and, therefore by equation (25), the corresponding spatial extent is larger and then more appropriate to cover the entire beam. The wavelet transforms are performed on the same displacement "eld but using di!erent mother wavelets, each of them using a di!erent relation between the parameter a and the acoustic frequency (equation (33)) and consequently, a di!erent design criteria a (equation (34)). The approximation using these multiple wavelet transforms then takes the form
L 2n K?G x PU?TCJCRQ+u w(x) t dx %G a a K G G G \ ?G
(35)
where a and t are associated with the ith wavelet transform, n is the number of wavelet G %G transforms employed and [!K, K] is the region outside which the mother wavelet has negligible values. The scale parameters are given in this case by c 2n a" i*, i"1, 2 , n, G u 2i!1
(36)
while the design criterion for each mother wavelet is given by (2i!1) a" , i"1, 2 , n. G 16n
(37)
For the ba%ed beam, the Morlet wavelet can thus be considered to be spatially con"ned and rewritten as a discrete sum, with b"0 and six displacement estimation points:
L 2n x H Dx , w t P U?TCJCRQ+u H % G a a G G G H
(38)
where the displacement values w are obtained from strain values using a "nite di!erences H scheme, for example. This approach is thus presented as an alternative to the Fourier
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P. MASSON AND A. BERRY
transform and the integral, as presented in equation (10), which have to be used for each frequency component. In fact, the proposed approximation of the radiated acoustic power only requires very few wavelet transforms to approximate the radiated acoustic power over the complete frequency range. Indeed, the resulting components are expressed in terms of the parameter a and a mapping to the frequency domain (u ) exists and is given by equation (36). The net computational gain obtained using the wavelet approximation over the Fourier approach is thus dependent on the number of frequency components where the radiated acoustic power is to be evaluated.
4. EXPERIMENTAL VALIDATION OF THE OPTIMIZED APPROACHES
The use of the optimized approaches is assessed through a comparison with the reference "nite di!erences approach using experimental measurements.
4.1.
EXPERIMENTAL SET-UP
The experimental set-up used for the validation consists of a simply supported ba%ed beam onto which, in addition to two piezoelectric patches, a set of PVDF sensors are bonded. The following sections describe the details of the experimental implementation. 4.1.1. Mechanical set-up A simply supported thin beam is mounted in a ba%e, as schematically presented in Figure 1. Four PVDF sensors and two piezoceramic (PZT) patches are glued to the beam and located as shown in Figure 2. The beam is excited at a single frequency by a HewlettPackard 3245A signal generator and a Bruel and Kjaer 4810 shaker between 60 Hz and 1 kHz to produce the primary disturbance vibratory "eld. The characteristics of the mechanical components are presented in Table 1. 4.1.2. Sensors and actuators The PVDF sensors are used to measure the strain at four discrete points. Each of these sensors is tailored with the same dimension, shown in Table 1. Considering the frequency range of interest and the modal behavior of the beam used in this work, the spatial extent of the PVDF sensors is small enough (as compared to structural wavelength) that a local strain measurement is obtained. A conductive adhesive transfer tape (3M) is used to bond the sensors to the beam. The quantity measured by the PVDF sources is in fact the integral
Figure 2. Experimental set-up: the sensors and the actuators.
717
ASAC STRATEGIES USING STRAIN MEASUREMENTS
TABLE 1 Components characteristics Beam Material
Aluminum
x dimension (m) y dimension (m) Young's modulus E (GPa) Thickness h (m) Poisson's ratio l Density o (kg/m) Piezoelectric strain constant d (m/V)
0)355 0)0252 67 0)003 0)3 2710 *
Sensors
Actuator(s)
PVDF AMP Inc., PZT Piezo Systems AMP 028 CN Inc., PSI-5A-S4-ENH 0)0035 0)0252 2 0)000028 * 3400 2)3;10\
0)025 0)0252 66 0)000254 0)29 7800 1)9;10\
of the strain over the surface of each sensor: q"!(h #h )l e @ .4"$ W
J
.4"$
e (x) dx, V
(39)
where l and h are the length and the thickness of the PVDF, respectively, and e is .4"$ ."4$ the piezoelectric stress constant (the piezoelectric stress constant e is related to the piezoelectric strain constant d by e"E d, where E is the Young's modulus of the N N piezoelectric). The charge signals q are converted to voltages using in-house charge ampli"ers. These signals are then "ltered by Rockland 852 anti-aliasing "lters at 1 kHz and a sampling frequency of 3 kHz is used. The actuators used for the control consist of two sets of colocated piezoceramics mounted on opposite faces of the beam and driven by a voltage such that the mechanical action of both piezoceramics is out of phase with respect to each other, i.e., one piezoceramic extends when the other contracts. It has been shown that in such a con"guration, the action of each piezoceramic patch is equivalent to the application of two moments at the edges of the patch [21]. Moreover, these moments are proportional to the voltage applied to the piezoceramic patch. The "rst piezoceramic patch is located at x"0)0075 m while the second patch is located at x"!0)0775 m. The control signals fed to the actuators are "ltered by a Rockland 2382 anti-aliasing "lter at 1 kHz and ampli"ed using a PCB Piezotronics AVC 790 high-voltage ampli"er. 4.1.3. Controller con,guration A feedforwad con"guration was used for the controller together with a "ltered-X least-mean-square (LMS) algorithm. The cost function used in the experimental validation is the radiated acoustic power approximated from structural measurements using the three previous approaches. In order to allow for the calculation of this cost function, the control algorithm is divided into two parts: the control part (real-time task) and the optimization part (idle processor time task), as shown in Figure 3. An o!-line adaptation of a correction to the control "lter is performed during the idle processor time and added to the control "lter when possible. In practice, for the cases studied in this paper, this update takes less than 0)5 s. FIR "lters are used for the control path model and the control "lter. The control path model is identi"ed o!-line using 1024 coe$cients while the control "lter uses four
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P. MASSON AND A. BERRY
Figure 3. O!-line controller con"guration.
Figure 4. Schematic representation of the control system.
coe$cients. The reference signal used by the feedforward controller is taken directly from the signal generator. Figure 4 presents the complete control system with the PVDF sensors and the PZT actuators. The controller operates on a Spectrum board equipped with a TMS 320C31 DSP. The National Instruments LabVIEW environment is used to download the code into the DSP and monitor the controller operation. 4.1.4. Control performance metrics A set of seven TMS ICP 130M01/P10 microphones located on a hemisphere in front of the beam as well as a Poltype OFV 2600-302 laser vibrometer are used to monitor the control performance of the approaches, through a Hewlett-Packard 3566A spectrum analyzer. In this respect, the radiated acoustic power is obtained from the pressure measurements by (equation (8)) 1 2nR P" " p (R, h , u ) ", G G G 2o c 7 G
(40)
ASAC STRATEGIES USING STRAIN MEASUREMENTS
719
where R"0)52 m, while the mean-square velocity 1wR 2 is obtained from nine equally spaced velocity measurements on the beam by 1 1wR 2" " wR "Dx. G 4l V G
(41)
The measurements were conducted in a semi-anechoic room having a cut-o! frequency of 100 Hz.
4.2.
EXPERIMENTAL RESULTS
Experimental results are presented to assess the proposed optimized approaches for the use of strain sensing in active structural acoustic control. The use of strain measurements for the estimation of radiated acoustic power is "rst experimentally assessed through the reference "nite di!erences approach. The control performance obtained using this estimation is also presented. The control performance of both optimized approaches is then presented and compared with the control performance obtained using the reference approach. A comparison of the "nite di!erences approach with an approach involving the minimization of the sum of the strain measurements has been presented before by the authors [22]. 4.2.1. Finite di+erences approach The "rst experimental validation concerns the ability of the "nite di!erences scheme to reconstruct the displacement "eld from the strain "eld and to provide an adequate estimate of the radiated acoustic power. The estimation of this radical acoustic power from velocity (and hence, displacement) measurements given by the laser vibrometer and from strain measurements is compared in Figure 5, together with the radiated acoustic power obtained from pressure measurements. This "gure shows that the reconstruction scheme tends to
Figure 5. Radiated acoustic power obtained from displacement "eld reconstruction from strain measurements and as calculated from pressure and velocity measurements: - - -, from pressure measurements; *, from velocity measurements; 0 , from strain measurements.
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P. MASSON AND A. BERRY
Figure 6. E!ect of the number of actuators for the control of the radiated acoustic power using the "nite di!erences approach: - - -, before control; **, control one actuator; 00, control two actuators.
underestimate the radiated acoustic power at low frequency, which can be explained by a poor signal-to-noise ratio involved in the "nite di!erences scheme in this region. At high frequency, an insu$cient number of sensors translates into a poor estimate of the radiated acoustic power. Control results are now presented in Figure 6 with the e!ect of the number of actuators used for the control in the "nite di!erences approach. It appears that an increase in the number of actuators is bene"cial to the control performance at low frequency. A possible explanation to this could come from the control objective which requires the restructuring of the vibration pattern instead of the direct attenuation of the vibration levels. 4.2.2. Direct approach All the results presented in the next two sections use two actuators for the control. The control performance of the direct approach is "rst compared to the control performance of the "nite di!erences approach. The radiated acoustic power obtained from pressure measurements after control with these approaches is presented in Figure 7. The performance of the direct approach is noticeable, considering that it involves a lower computational burden. This could translate into a faster adaptation to variations in the statistics of the disturbance signal. The e!ect of both approaches on the mean-square velocity is presented in Figure 8, which reveals that the direct approach aims more at minimizing the vibrations of the structure. This is more apparent on the radiation e$ciency p, de"ned as P p" 4o cl l 1wR 2 VW which is presented in Figure 9.
(42)
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Figure 7. Comparison of the radiated acoustic power as controlled using the "nite di!erences approach and the direct approach: - - -, before control: **, "nite di!erences: 00, direct approach. Two actuators are used.
Figure 8. Comparison of the mean-square velocity as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the direct approach: - - -, before control: **, "nite di!erences: 00, direct approach. Two actuators are used.
The reconstruction problem noticed at low frequency in the "nite di!erences approach (see Figure 5) is further illustrated in Figure 10 which shows the cost function in the direct and "nite di!erences approaches after the control has been applied. The curves presented in this "gure show that the "nite di!erences approach provides a biased estimate of the radiated acoustic power, especially at low frequency, which results in a slightly smaller acoustic power reduction.
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Figure 9. Comparison of the radiation e$ciency as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the direct approach: - - -, before control; **, "nite di!erences; 00, direct approach. Two actuators are used.
Figure 10. Comparison of the radiated acoustic power as measured and as obtained from the cost function after the control using the "nite di!erences approach, the direct approach and the wavelet approach: - - -, from pressure measurements; ** cost function "nite di!erences: 00, cost function direct approach; - - -, cost function wavelet approach. Two actuators are used.
4.2.3. =avelet approach The second optimized approach to be compared to the "nite di!erences approach is the wavelet approach. The wavelet approach is of interest because it allows a fast approximation of the radiated acoustic power without making, by itself, any assumptions on the boundary conditions. For these experimental results, n"4 wavelet transforms are
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723
Figure 11. Comparison of the radiated acoustic power as controlled using the "nite di!erences approach and the wavelet approach: - - -, before control: **, "nite di!erences; 00, wavelet approach. Two actuators are used.
Figure 12. Comparison of the mean-square velocity as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the wavelet approach: - - -, before approach: **, "nite di!erences; 00, wavelet approach. Two actuators are used.
used to estimate the radiated acoustic power. The comparison is presented in Figures 11}13. The good control performance of the wavelet approach is very encouraging since it could provide a very e$cient tool for the broadband control of the acoustic power radiated from structures. These results con"rm the optimal control results obtained previously [15]. Moreover, as seen on the mean-square velocity and on the radiation e$ciency, the control mechanism involved in the wavelet approach seems to resemble the control mechanism
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P. MASSON AND A. BERRY
Figure 13. Comparison of the radiation e$ciency as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the wavelet approach: - - - , before control; **, "nite di!erences; 00, wavelet approach. Two actuators are used.
involved in the radiated acoustic power control as estimated by the "nite di!erences scheme. It should be noted that the wavelet approach also su!ers from reconstruction problems at low frequency (see section 4.2.1) as it is based on the use of a reconstructed displacement "eld; this is illustrated in Figure 10.
5. CONCLUSIONS
Two optimized approaches were presented and experimentally validated using piezoelectric discrete strain sensors. A ba%ed simply supported beam was used for the validation. The "rst optimized approach was presented to allow the evaluation of the radiated acoustic power directly from the strain measurements, without the need for a reconstruction scheme. The second optimized approach was presented to accelerate the evaluation of the radiated acoustic power through an estimate based on the wavelet transform. The experimental set-up was presented, together with some details on the real-time implementation of the approaches. Pressure and velocity measurements were made to better understand and illustrate the control mechanisms involved in the di!erent approaches. The experimental results show that the reference "nite di!erences approach su!ers from a poor signal-to-noise ratio at low frequency, but still o!ers a very interesting control performance over the whole frequency range. These results con"rm previous analytical results indicating a reconstruction problem at low frequency. The results also show that the optimized approaches perform very well in attending the radiated acoustic power under control. The direct approach seems to provide a more realistic representation of the radiated acoustic power to the controller. This approach is however restricted to cases where boundary terms can be neglected. Being applicable to more general structures, the wavelet approach represents an interesting alternative as it appears to behave like the
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complete radiated acoustic power approach. This indicates that this approach is expected to be more e$cient for planar structures especially if broadband control is required. Fast algorithms involving "lter banks for the wavelet transform can be used. Because they use discrete sensors, all the approaches can be extended to two-dimensional structures, preserving the distinction between the wavenumber components, still allowing for a wavenumber approach. Work on the extension of the approaches to two-dimensional structures with di!erent boundary conditions is currently in progress. The exploitation of spatial localization property of the wavelet transform is also being investigated.
ACKNOWLEDGMENT
This work was supported by the Natural Sciences and Engineering Research Council (NSERC), Canada.
REFERENCES 1. S. J. ELLIOTT and P. A. NELSON 1993 IEEE Signal Processing Magazine 10, 12}35. Active noise control. 2. C. R. FULLER 1990 Journal of Sound and
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17. R. L. CLARK and C. R. FULLER 1991 Proceedings of the Conference on Recent Advances in Active Control of Sound and
COMPARISON OF SEVERAL STRATEGIES IN THE ACTIVE STRUCTURAL ACOUSTIC CONTROL USING STRUCTURAL STRAIN MEASUREMENTS P. MASSON AND A. BERRY G.A.;.S, Mechanical Engineering Department, ;niversiteH de Sherbrooke, Sherbrooke, QueH bec, Canada J1K 2R1 (Received 21 January 1999, and in ,nal form 16 December 1999) Two optimized approaches are herein proposed to lighten the real-time computation of the radiated acoustic power from planar structures using discrete strain measurements. These approaches are compared to a reference approach involving "nite di!erences reconstruction of the displacement "eld from the strain "eld. The "rst optimized approach allows the direct calculation of the radiated acoustic power from the discrete strain "eld while the second optimized approach employs an accelerated evaluation of the radiated acoustic power based on a wavelet transform of the displacement "eld. The evaluation of the radiated acoustic power from discrete strain values is experimentally assessed using the reference "nite di!erences approach and both optimized approaches. Piezoelectric "lm (PVDF) strain sensors are used for this validation as well as a feedforward "ltered-X LMS controller. Details on the real-time implementation of the cost functions are presented, especially on the o!-line adaptation of the control "lter. Formal comparisons using acoustic and vibration measurements are presented to better illustrate the control performance and mechanism of the di!erent approaches. Experimental results serve to demonstrate the interest of using the optimized approaches as compared to the reference approach and it is shown that a signi"cant gain in computational burden can be obtained using the proposed optimized approaches without sacri"cing the control performance. 2000 Academic Press 1. INTRODUCTION
The development of high-performance dedicated signal processors has driven the interest for active noise control in recent years. These processors are now more compact and a variety of tools exists to easily implement numerical controllers on them. It is well known that passive methods involving foams, viscoelastic layers and other absorbing or dissipative materials cannot be used e$ciently to reduce low-frequency noise. To perform noise control at those frequencies, di!erent active control strategies have been proposed and can be distinguished as active noise control (ANC) [1] and active structural acoustic control (ASAC) [2]. While active noise control requires the use of actuators located in the acoustic "eld, i.e., loudspeakers, the active structural acoustic control allows for the integration of both the error sensors and the actuators in the structure itself, following the trend towards smart materials. Moreover, it has been shown that, most of the time, ASAC has a lower requirement on the number of secondary sources needed to perform the control as compared to ANC [1]. Among the active structural acoustic control strategies, global and local strategies can be distinguished. Local control refers to spatially localized control (zones of silence) while global control is associated with extended control e!ect using cost functions closely related 0022-460X/00/240707#20 $35.00/0
2000 Academic Press
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to energy quantities [3]. In this respect, the information given by the error sensors was used in the de"nition of cost functions based, for example, either on the far"eld pressure in prescribed directions [4}6], the volume velocity [7, 8] or the radiated acoustic power [9, 10]. The latter appears to be very promising as it ensures global control and, moreover, if working in the wavenumber domain [11, 12], allows for proper concentration of the control e!ort in the supersonic region of the structural displacement wavenumber spectrum, where the radiating wavenumber components are located. The control strategy based on the radiated acoustic power and de"ned in the wavenumber domain "rst requires the wavenumber transformation of the displacement (or acceleration) "eld of the structure. The supersonic (radiating) components of the spectrum can then be identi"ed and minimized. For this purpose, as compared to spatially extended sensors [13, 14], discrete sensors are more appropriate as they allow to distinguish between the structural components on two-dimensional structures [4]. Moreover, following the trend towards smart materials, strain sensors appear to be well suited for integration into structures and materials. As they do not measure directly the displacement or acceleration of the structure, the use of these sensors implies an additional step for the estimation of the radiated acoustic power. Therefore, there is a need to reduce the computational burden associated with the use of such sensors. This paper is thus concerned with the experimental validation of two optimized approaches which are herein proposed to evaluate the radiated acoustic power from discrete strain measurements. The cost function is de"ned as global and is expressed as the radiated acoustic power in the wavenumber domain and using discrete strain sensors. The "rst optimized approach allows the calculation of the radiated acoustic power directly from the strain "eld, without requiring any reconstruction of the displacement "eld. This "rst approach is based on a double integration by parts of the radiated acoustic power integral to provide an estimate in terms of the wavenumber transform of the strain "eld. The second optimized approach is directed towards the acceleration of the radiated acoustic power calculation by using a wavelet transform of a reconstructed displacement "eld, thus replacing both the Fourier transform and the integration of the radiating components [15]. In this second approach, the wavelet transform is chosen because it can have e$cient restriction on the supersonic region using a single space-wavenumber transformation. It is known that the wavelet transform has the property to localize the energy both in a given region of the wavenumber spectrum and spatially on the structure. The approach presented in this paper thus bene"ts from the wavelet framework, even if the analysis is herein restricted to the wavenumber localization in order to simplify the illustration of the approach. The reference approach based on a "nite di!erences reconstruction of the displacement "eld is "rst brie#y presented. Both optimized approaches are then presented and their control performance is experimentally compared to the reference approach.
2. ACOUSTIC RADIATION IN THE WAVENUMBER DOMAIN
For the sake of illustration, the acoustic radiation from a ba%ed rectangular thin beam is considered in this work. The acoustic pressure p(M) radiated at a point M from a ba%ed beam can be expressed by the Rayleigh's integral: o p(M)" 2n
=G (P, t)
e\ IP dA, r
(1)
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Figure 1. Experimental set-up: the ba%ed beam.
G (P, t) is the beam transverse acceleration at where o is the mass density of the #uid, = a point P, k is the acoustic wavenumber in the #uid, r is the distance between P and M and A is the surface of the beam, as presented in Figure 1. Assuming harmonic and constant displacement along the y direction such that =(P, t)"w(x) e SR, the previous equation can be used to obtain a far"eld approximation of the radiated acoustic pressure [16]: u o e\ I0 p(R, , u)"! wJ (j, k), 2n R
(2)
where j"k sin h cos u,
k"k sin h sin u
(3, 4)
and where R, h and u are the spherical co-ordinates of the point M. wJ (j, k) is the double wavenumber transform of the beam displacement, analogous to the inverse Fourier transform de"ned, for a beam of "nite width assuming constant displacement along the y direction, as [7] JV
JW
\J \J w(x, y) e I V>I W dy dx
wJ (k , k )" V W
V
V
W
(5)
W
2 sin k l JV WW " w(x) e\ IVV dx k W \JV
(6)
2 sin k l W W wJ (k ), " V k W
(7)
where l is the half-length of the beam, l is the halfwidth of the beam and k and k are, V W V W respectively, the x and y components of the structural wavenumber.
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The acoustic power P radiated from the beam can be expressed as the integral over a hemisphere of the mean square of the sound pressure in the far "eld:
1 P" " p(R, h, u) " dS, 2o c 1
(8)
where S is the surface of the hemisphere and c is the speed of sound in the #uid. In order to exploit the physical mechanisms of structural radiation, the radiated acoustic power is expressed in the wavenumber domain [11, 17, 14]. The radiated acoustic power in the wavenumber domain is obtained by substituting equation (2) into equation (8), as presented in detail in reference [18]: o u I (I\H " wJ (j, k) " dk dj. P" 8n \I \(I\H (k!j!k
(9)
In equation (9), the integration is performed over the supersonic region of the structural wavenumber spectrum (i.e., the region where the acoustic wavenumber k is larger than the structural wavenumber given by (j#k), which shows that only supersonic structural waves radiate sound in the far "eld. The minimization of the displacement wavenumber spectrum (weighted by (1/(k!j!k) in the supersonic region will then minimize the radiated acoustic power. For a narrow beam (for which " k l " 1) and using equations (2) and (7), equation (9) WW reduces to the form o u l I P " W " wJ (j) " dj. DGLGRC 2n \I
(10)
The objective of the following control approaches is to derive cost functions from structural strain measurements which approximate the exact radiated acoustic power given by equation (10).
3. APPROACHES FOR USING STRAIN MEASUREMENTS IN THE EVALUATION OF THE RADIATED ACOUSTIC POWER
Three approaches are herein presented for using strain measurements in the evaluation of the radiated acoustic power, through equation (10). The "rst approach, based on a "nite di!erences scheme for the reconstruction of the displacement "eld from the strain "eld, is referred to as the reference approach because it involves fewer assumptions than other approaches. Two optimized approaches are also presented: the "rst one is based on rewriting equation (10) in terms of the wavenumber transform of the strain "eld; the second one is based on a wavelet approximation of the radiated acoustic power. In the following, it will be assumed that the Bernoulli thin beam conditions apply, i.e., the strain e (x) at the surface of the beam is related to the second space derivative, in that V direction, of the displacement "eld w(x): h *w(x) e (x)"! @ , V 2 *x where h is the thickness of the beam. @
(11)
ASAC STRATEGIES USING STRAIN MEASUREMENTS
3.1.
711
REFERENCE APPROACH: FINITE DIFFERENCES SCHEME
A "nite di!erences scheme can be used to reconstruct the displacement "eld w(x) from the discrete strain "eld *w(x)/*x known on a p equally spaced grid of points on the beam (x ) G (i"1, 2, 2 , p, where i"1 and i"p are on edges) [10]. This scheme however requires the knowledge of two of the boundary conditions of the beam. The central-di!erence scheme can be written in a matrix form as A
w "w , N N> N> N
(12)
where the subscripts indicate the dimensions of the matrices or vectors. In this equation, w is the vector of the measured strain and w is built from the displacement at each N N> measurement point w and from the slopes of the displacement "eld in the x direction G w ,*w(x )/*x on the edges such that H H w w w w " . N> $
(13)
w N w N Indeed, it is seen that equtaion (12) can be solved only if two components of w are N> known a priori. These two components can be chosen from the "rst two components and the last two components of this vector, representing boundary conditions on the displacement or the slope along the x"x and x"x edges. In the following, four physical N strain measurement points are used, in addition to two assumed strain values on the edges. Under these conditions and for a simply supported beam, taking into account the zero-strain condition (w "0) as well as the zero-displacement condition (w"0) on the edges, the "nite di!erences scheme reduces to !2Dx
2 !2
1 Dx
1
1 !2
1
1 !2
1
1 !2 2 2Dx
w 0 w w w w " , w w w w w 0
(14)
where Dx is the space interval between two consecutive points and w "*w(x )/*x is the I I strain measured in the x direction. The reconstructed displacement "eld obtained from the vector w , by solving equation (14), and from the zero-displacement assumption on the edges, is then transformed using a discretized version of equation (6) given by wJ (k )" w e IV VGDx V G and substituted in equation (10) to obtain the radiated acoustic power.
(15)
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In order to obey the Nyquist criterion, it is expected that two measurement points will be required per mechanical wavelength to obtain an adequate estimate of the radiated acoustic power. Therefore, four strain sensors along the beam, together with the two assumed zero-strain values at the beam edges, should allow an adequate estimate up to the third vibration mode. Spatial aliasing can impair the estimate of the radiated acoustic power as the precision obtained on the wavenumber components of the displacement "eld is directly related to the accuracy of the radiated acoustic power estimate. The use of zero padding can improve the accuracy of the estimate by providing a larger number of components in the evaluation of the radiation integral given in equation (10). Spatial aliasing was extensively studied before through numerical simulations [10].
3.2.
OPTIMIZED APPROACHES
3.2.1. Direct approach The "rst optimized approach aims at expressing the radiated acoustic power (equation (10)) directly in terms of the Fourier transform of the strain measurements. This approach is based on a double integration by parts of the wavenumber transform (equation (5)). The "rst integration by parts with respect to the x variable leads to the following result for the wavenumber transform: 1 wJ (k , k )" V W ik X
1 ? @ *w(x, y) w(x, y)e IV V>IW Wn ) i dl! e IVV>IW W dy dx, ik *x V \? \@
(16)
where i is the unit vector in the x direction, n is the unit outer vector normal to an edge and Z denotes the integration over the contour of the surface. The second integration by parts with respect to the x variable gives
1 1 w(x, y) e IV V>IW Wn ) i dl# wJ (k , k )" V W k ik X V
*w(x, y) e IV V>IW W n ) i dl *x
1 ! SI (k , k ) V W k V
(17)
with ?
@ *w(x, y) e IV V>IW W dy dx, *x \? \@
SI (k , k )" V W
(18)
where SI (k , k ) is the wavenumber transform of the strain "eld. Equation (17) can be V W rewritten as 1 wJ (k , k )" I ! SI (k , k ), V W G k V W V G
(19)
where I are the subsisting boundary terms associated with edges i"1, 2 parallel to the G y-axis. Preceding work has shown that, for a simply supported structure, the boundary terms can be neglected at low frequency without a!ecting signi"cantly the radiated acoustic
ASAC STRATEGIES USING STRAIN MEASUREMENTS
713
power estimate [10]. It was observed that, for a given mode, the boundary term in equation (19) and the surface strain transform term in equation (18) are of the same order of magnitude, below the modal frequency. The importance of the surface strain-transform term increases slightly close to that frequency while its relative importance decreases rapidly above this frequency. The global e!ect is that, with an increase in frequency, more and more modes are considered in which the surface-strain transform term has little e!ect or, conversely, the boundary term has large relative e!ect. Therefore, for a simply supported beam and using four strain sensors, the following one-dimensional form of equation (19) can be used if k O0: V 1 wJ (k )+! w eGIVVG Dx, V G k V G
(20)
and if k "0, the following form is used [10]: V
x l G ! V w Dx. wJ (0)+ G 2 2 G
(21)
3.2.2. =avelet approach In the "nite di!erences approach, the radiated acoustic power was related to the integration, in the supersonic region of the wavenumber domain, of the squared norm of the spectral components (corresponding to energy density) of the displacement "eld, as it appears in equation (10). The second optimized approach uses a space-wavenumber wavelet transform to observe the energy localized in the supersonic region. This approach has been theoretically investigated before [15] and a condensed presentation of the approach is only given here. The wavelet transform of a function f (x) is de"ned as
1 x!b f (x) t (= f ) (b, a)" dx, R a (" a " \
(22)
where t(x) is called the mother wavelet from which all the other wavelets are generated using a translation parameter b and a dilation parameter a. The overbar indicates the complex conjugate. It can be shown [19] that this wavelet transform gives localized information on a function f (x) within the space and wavenumber windows which can be de"ned in the root-mean-square (r.m.s.) sense, consequently leading to a localization which is also expressed in the mean-square sense. The information is then localized with the wavelet transform within a space window given by [b#ax*!aD , b#ax*#aD ], R R
(23)
where x* is the center and D is the r.m.s. extent (or radius) of the mother wavelet. In R a similar way, the wavelet transform is seen to give localized information in the wavenumber domain within a wavenumber window given by
i* D i* D ! R, # R , a a a a
(24)
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P. MASSON AND A. BERRY
where tI (i) is the Fourier transform of the mother wavelet and, as before, i* is the center and D I is the r.m.s. bandwidth (or radius) of the Fourier transform of this mother wavelet. R The quantities D and D I also de"nes space and wavenumber resolutions, associated with R R a particular wavelet. These resolutions satisfy the uncertainty principle [19]: D D I * . R R
(25)
The squared norm of the Fourier transform given by " fI (i) " is known to lead to a continuous representation of the energy density distribution of f (x) in the wavenumber domain, known as a spectrogram. The energy density is thus obtained only as a function of i. To obtain an energy density distribution of the function in the space}wavenumber plane, a similar quantity can be de"ned in the case of the wavelet transform as a squared norm given by " (= f ) (b, a) " R
(26)
and this quantity can be referred to as a scalogram [20]. The resolution capabilities of the scalogram depend on the analyzed wavenumber, as presented before. The shape of the wavelet t and of its Fourier tI obviously determines a weighting for the localization obtained in both domains. Following equations (23) and (24), the energy is distributed into space and wavenumber windows in the mean-square sense depending on the parameters b and a. Considering the above, it is "rst proposed in this section that the squared norm of the wavelet transform of the beam-displacement "eld, " (= w) (b, a) ", can observe the R contribution of this local displacement "eld to the total vibratory energy [15]. It is next proposed that the quantity given by
2n (= w) (b, a) u (" a " R
(27)
can be used to represent the acoustic power radiated by the beam if the parameter a is chosen such that the wavelet transform gives localized energy information in the supersonic region of the wavenumber spectrum. In other words, the choice of the parameter a as well as the use of the multiplying factor u are such that equation (27) mimics the exact expression of the radiated power (10). It is further assumed that a single component in the b-axis is su$cient to represent the energy in the supersonic region. Therefore, the parameter b is set to 0, implying that the wavelet is always centered on the beam. In the following, due to its simplicity, the Morlet wavelet will be chosen as the mother wavelet even if, rigorously, this wavelet is only numerically admissible, i.e., only numerical reconstruction of f (x) from its wavelet transform is possible. The Morlet wavelet t (x) is % de"ned as the modulated Gaussian window is given by 1 t (x) " : e\ LV> e\V ?, % 2(na
(28)
where a will be chosen in order to meet special requirements, as shown later. The following centers and radii are easily obtained for the Morlet wavelet: x*"0,
D "(a, R
i*"!2n,
1 DI" . R 2(a
(29}32)
ASAC STRATEGIES USING STRAIN MEASUREMENTS
715
It has been shown [15] that the following relation between the parameter a and the angular frequency u allows the wavelet transform to localize the energy in the supersonic region of the wavenumber spectrum at frequency u : c i* a" , u c
(33)
where a parameter 0)5)c)1 is used to ensure that, considering the center speci"ed for the window, its width will not cause the window to exceed the supersonic region. Therefore, a design criterion for the Morlet wavelet can be obtained under these conditions: 1 c a" . 16n (1!c)
(34)
In the preceding, the wavelet approach was presented using one-wavelet transform. However, in order to cover completely both space and wavenumber domains, i.e., the entire beam and the entire supersonic region, a few wavelet transforms have to be used and thus, a few wavelets in the wavenumber domain, each of them having b"0 and each of them reconstructing the energy in a given subregion of the supersonic region, and then summing all these contributions. By doing so, the wavenumber window of each wavelet is of limited extent and, therefore by equation (25), the corresponding spatial extent is larger and then more appropriate to cover the entire beam. The wavelet transforms are performed on the same displacement "eld but using di!erent mother wavelets, each of them using a di!erent relation between the parameter a and the acoustic frequency (equation (33)) and consequently, a di!erent design criteria a (equation (34)). The approximation using these multiple wavelet transforms then takes the form
L 2n K?G x PU?TCJCRQ+u w(x) t dx %G a a K G G G \ ?G
(35)
where a and t are associated with the ith wavelet transform, n is the number of wavelet G %G transforms employed and [!K, K] is the region outside which the mother wavelet has negligible values. The scale parameters are given in this case by c 2n a" i*, i"1, 2 , n, G u 2i!1
(36)
while the design criterion for each mother wavelet is given by (2i!1) a" , i"1, 2 , n. G 16n
(37)
For the ba%ed beam, the Morlet wavelet can thus be considered to be spatially con"ned and rewritten as a discrete sum, with b"0 and six displacement estimation points:
L 2n x H Dx , w t P U?TCJCRQ+u H % G a a G G G H
(38)
where the displacement values w are obtained from strain values using a "nite di!erences H scheme, for example. This approach is thus presented as an alternative to the Fourier
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transform and the integral, as presented in equation (10), which have to be used for each frequency component. In fact, the proposed approximation of the radiated acoustic power only requires very few wavelet transforms to approximate the radiated acoustic power over the complete frequency range. Indeed, the resulting components are expressed in terms of the parameter a and a mapping to the frequency domain (u ) exists and is given by equation (36). The net computational gain obtained using the wavelet approximation over the Fourier approach is thus dependent on the number of frequency components where the radiated acoustic power is to be evaluated.
4. EXPERIMENTAL VALIDATION OF THE OPTIMIZED APPROACHES
The use of the optimized approaches is assessed through a comparison with the reference "nite di!erences approach using experimental measurements.
4.1.
EXPERIMENTAL SET-UP
The experimental set-up used for the validation consists of a simply supported ba%ed beam onto which, in addition to two piezoelectric patches, a set of PVDF sensors are bonded. The following sections describe the details of the experimental implementation. 4.1.1. Mechanical set-up A simply supported thin beam is mounted in a ba%e, as schematically presented in Figure 1. Four PVDF sensors and two piezoceramic (PZT) patches are glued to the beam and located as shown in Figure 2. The beam is excited at a single frequency by a HewlettPackard 3245A signal generator and a Bruel and Kjaer 4810 shaker between 60 Hz and 1 kHz to produce the primary disturbance vibratory "eld. The characteristics of the mechanical components are presented in Table 1. 4.1.2. Sensors and actuators The PVDF sensors are used to measure the strain at four discrete points. Each of these sensors is tailored with the same dimension, shown in Table 1. Considering the frequency range of interest and the modal behavior of the beam used in this work, the spatial extent of the PVDF sensors is small enough (as compared to structural wavelength) that a local strain measurement is obtained. A conductive adhesive transfer tape (3M) is used to bond the sensors to the beam. The quantity measured by the PVDF sources is in fact the integral
Figure 2. Experimental set-up: the sensors and the actuators.
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ASAC STRATEGIES USING STRAIN MEASUREMENTS
TABLE 1 Components characteristics Beam Material
Aluminum
x dimension (m) y dimension (m) Young's modulus E (GPa) Thickness h (m) Poisson's ratio l Density o (kg/m) Piezoelectric strain constant d (m/V)
0)355 0)0252 67 0)003 0)3 2710 *
Sensors
Actuator(s)
PVDF AMP Inc., PZT Piezo Systems AMP 028 CN Inc., PSI-5A-S4-ENH 0)0035 0)0252 2 0)000028 * 3400 2)3;10\
0)025 0)0252 66 0)000254 0)29 7800 1)9;10\
of the strain over the surface of each sensor: q"!(h #h )l e @ .4"$ W
J
.4"$
e (x) dx, V
(39)
where l and h are the length and the thickness of the PVDF, respectively, and e is .4"$ ."4$ the piezoelectric stress constant (the piezoelectric stress constant e is related to the piezoelectric strain constant d by e"E d, where E is the Young's modulus of the N N piezoelectric). The charge signals q are converted to voltages using in-house charge ampli"ers. These signals are then "ltered by Rockland 852 anti-aliasing "lters at 1 kHz and a sampling frequency of 3 kHz is used. The actuators used for the control consist of two sets of colocated piezoceramics mounted on opposite faces of the beam and driven by a voltage such that the mechanical action of both piezoceramics is out of phase with respect to each other, i.e., one piezoceramic extends when the other contracts. It has been shown that in such a con"guration, the action of each piezoceramic patch is equivalent to the application of two moments at the edges of the patch [21]. Moreover, these moments are proportional to the voltage applied to the piezoceramic patch. The "rst piezoceramic patch is located at x"0)0075 m while the second patch is located at x"!0)0775 m. The control signals fed to the actuators are "ltered by a Rockland 2382 anti-aliasing "lter at 1 kHz and ampli"ed using a PCB Piezotronics AVC 790 high-voltage ampli"er. 4.1.3. Controller con,guration A feedforwad con"guration was used for the controller together with a "ltered-X least-mean-square (LMS) algorithm. The cost function used in the experimental validation is the radiated acoustic power approximated from structural measurements using the three previous approaches. In order to allow for the calculation of this cost function, the control algorithm is divided into two parts: the control part (real-time task) and the optimization part (idle processor time task), as shown in Figure 3. An o!-line adaptation of a correction to the control "lter is performed during the idle processor time and added to the control "lter when possible. In practice, for the cases studied in this paper, this update takes less than 0)5 s. FIR "lters are used for the control path model and the control "lter. The control path model is identi"ed o!-line using 1024 coe$cients while the control "lter uses four
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Figure 3. O!-line controller con"guration.
Figure 4. Schematic representation of the control system.
coe$cients. The reference signal used by the feedforward controller is taken directly from the signal generator. Figure 4 presents the complete control system with the PVDF sensors and the PZT actuators. The controller operates on a Spectrum board equipped with a TMS 320C31 DSP. The National Instruments LabVIEW environment is used to download the code into the DSP and monitor the controller operation. 4.1.4. Control performance metrics A set of seven TMS ICP 130M01/P10 microphones located on a hemisphere in front of the beam as well as a Poltype OFV 2600-302 laser vibrometer are used to monitor the control performance of the approaches, through a Hewlett-Packard 3566A spectrum analyzer. In this respect, the radiated acoustic power is obtained from the pressure measurements by (equation (8)) 1 2nR P" " p (R, h , u ) ", G G G 2o c 7 G
(40)
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where R"0)52 m, while the mean-square velocity 1wR 2 is obtained from nine equally spaced velocity measurements on the beam by 1 1wR 2" " wR "Dx. G 4l V G
(41)
The measurements were conducted in a semi-anechoic room having a cut-o! frequency of 100 Hz.
4.2.
EXPERIMENTAL RESULTS
Experimental results are presented to assess the proposed optimized approaches for the use of strain sensing in active structural acoustic control. The use of strain measurements for the estimation of radiated acoustic power is "rst experimentally assessed through the reference "nite di!erences approach. The control performance obtained using this estimation is also presented. The control performance of both optimized approaches is then presented and compared with the control performance obtained using the reference approach. A comparison of the "nite di!erences approach with an approach involving the minimization of the sum of the strain measurements has been presented before by the authors [22]. 4.2.1. Finite di+erences approach The "rst experimental validation concerns the ability of the "nite di!erences scheme to reconstruct the displacement "eld from the strain "eld and to provide an adequate estimate of the radiated acoustic power. The estimation of this radical acoustic power from velocity (and hence, displacement) measurements given by the laser vibrometer and from strain measurements is compared in Figure 5, together with the radiated acoustic power obtained from pressure measurements. This "gure shows that the reconstruction scheme tends to
Figure 5. Radiated acoustic power obtained from displacement "eld reconstruction from strain measurements and as calculated from pressure and velocity measurements: - - -, from pressure measurements; *, from velocity measurements; 0 , from strain measurements.
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Figure 6. E!ect of the number of actuators for the control of the radiated acoustic power using the "nite di!erences approach: - - -, before control; **, control one actuator; 00, control two actuators.
underestimate the radiated acoustic power at low frequency, which can be explained by a poor signal-to-noise ratio involved in the "nite di!erences scheme in this region. At high frequency, an insu$cient number of sensors translates into a poor estimate of the radiated acoustic power. Control results are now presented in Figure 6 with the e!ect of the number of actuators used for the control in the "nite di!erences approach. It appears that an increase in the number of actuators is bene"cial to the control performance at low frequency. A possible explanation to this could come from the control objective which requires the restructuring of the vibration pattern instead of the direct attenuation of the vibration levels. 4.2.2. Direct approach All the results presented in the next two sections use two actuators for the control. The control performance of the direct approach is "rst compared to the control performance of the "nite di!erences approach. The radiated acoustic power obtained from pressure measurements after control with these approaches is presented in Figure 7. The performance of the direct approach is noticeable, considering that it involves a lower computational burden. This could translate into a faster adaptation to variations in the statistics of the disturbance signal. The e!ect of both approaches on the mean-square velocity is presented in Figure 8, which reveals that the direct approach aims more at minimizing the vibrations of the structure. This is more apparent on the radiation e$ciency p, de"ned as P p" 4o cl l 1wR 2 VW which is presented in Figure 9.
(42)
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Figure 7. Comparison of the radiated acoustic power as controlled using the "nite di!erences approach and the direct approach: - - -, before control: **, "nite di!erences: 00, direct approach. Two actuators are used.
Figure 8. Comparison of the mean-square velocity as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the direct approach: - - -, before control: **, "nite di!erences: 00, direct approach. Two actuators are used.
The reconstruction problem noticed at low frequency in the "nite di!erences approach (see Figure 5) is further illustrated in Figure 10 which shows the cost function in the direct and "nite di!erences approaches after the control has been applied. The curves presented in this "gure show that the "nite di!erences approach provides a biased estimate of the radiated acoustic power, especially at low frequency, which results in a slightly smaller acoustic power reduction.
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Figure 9. Comparison of the radiation e$ciency as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the direct approach: - - -, before control; **, "nite di!erences; 00, direct approach. Two actuators are used.
Figure 10. Comparison of the radiated acoustic power as measured and as obtained from the cost function after the control using the "nite di!erences approach, the direct approach and the wavelet approach: - - -, from pressure measurements; ** cost function "nite di!erences: 00, cost function direct approach; - - -, cost function wavelet approach. Two actuators are used.
4.2.3. =avelet approach The second optimized approach to be compared to the "nite di!erences approach is the wavelet approach. The wavelet approach is of interest because it allows a fast approximation of the radiated acoustic power without making, by itself, any assumptions on the boundary conditions. For these experimental results, n"4 wavelet transforms are
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Figure 11. Comparison of the radiated acoustic power as controlled using the "nite di!erences approach and the wavelet approach: - - -, before control: **, "nite di!erences; 00, wavelet approach. Two actuators are used.
Figure 12. Comparison of the mean-square velocity as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the wavelet approach: - - -, before approach: **, "nite di!erences; 00, wavelet approach. Two actuators are used.
used to estimate the radiated acoustic power. The comparison is presented in Figures 11}13. The good control performance of the wavelet approach is very encouraging since it could provide a very e$cient tool for the broadband control of the acoustic power radiated from structures. These results con"rm the optimal control results obtained previously [15]. Moreover, as seen on the mean-square velocity and on the radiation e$ciency, the control mechanism involved in the wavelet approach seems to resemble the control mechanism
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Figure 13. Comparison of the radiation e$ciency as obtained after the control of the radiated acoustic power using the "nite di!erences approach and the wavelet approach: - - - , before control; **, "nite di!erences; 00, wavelet approach. Two actuators are used.
involved in the radiated acoustic power control as estimated by the "nite di!erences scheme. It should be noted that the wavelet approach also su!ers from reconstruction problems at low frequency (see section 4.2.1) as it is based on the use of a reconstructed displacement "eld; this is illustrated in Figure 10.
5. CONCLUSIONS
Two optimized approaches were presented and experimentally validated using piezoelectric discrete strain sensors. A ba%ed simply supported beam was used for the validation. The "rst optimized approach was presented to allow the evaluation of the radiated acoustic power directly from the strain measurements, without the need for a reconstruction scheme. The second optimized approach was presented to accelerate the evaluation of the radiated acoustic power through an estimate based on the wavelet transform. The experimental set-up was presented, together with some details on the real-time implementation of the approaches. Pressure and velocity measurements were made to better understand and illustrate the control mechanisms involved in the di!erent approaches. The experimental results show that the reference "nite di!erences approach su!ers from a poor signal-to-noise ratio at low frequency, but still o!ers a very interesting control performance over the whole frequency range. These results con"rm previous analytical results indicating a reconstruction problem at low frequency. The results also show that the optimized approaches perform very well in attending the radiated acoustic power under control. The direct approach seems to provide a more realistic representation of the radiated acoustic power to the controller. This approach is however restricted to cases where boundary terms can be neglected. Being applicable to more general structures, the wavelet approach represents an interesting alternative as it appears to behave like the
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complete radiated acoustic power approach. This indicates that this approach is expected to be more e$cient for planar structures especially if broadband control is required. Fast algorithms involving "lter banks for the wavelet transform can be used. Because they use discrete sensors, all the approaches can be extended to two-dimensional structures, preserving the distinction between the wavenumber components, still allowing for a wavenumber approach. Work on the extension of the approaches to two-dimensional structures with di!erent boundary conditions is currently in progress. The exploitation of spatial localization property of the wavelet transform is also being investigated.
ACKNOWLEDGMENT
This work was supported by the Natural Sciences and Engineering Research Council (NSERC), Canada.
REFERENCES 1. S. J. ELLIOTT and P. A. NELSON 1993 IEEE Signal Processing Magazine 10, 12}35. Active noise control. 2. C. R. FULLER 1990 Journal of Sound and
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17. R. L. CLARK and C. R. FULLER 1991 Proceedings of the Conference on Recent Advances in Active Control of Sound and